Some people would use them interchangeably.
I think subject is more often used for people, or animate things, things that act. So you observe a man running, or a cat licking its paws; and the man or the cat is the subject of your observation.
"The cat was the subject of my attention."
I would use object for a rock, a painting, any inanimate thing.
"The painting was the sole object of my attention."
Someone else might have a different idea.